A math prediction that stood for decades was just proven wrong in higher dimensions.
April 17, 2026
Original Paper
Erdős's diameter conjecture for separated distances fails in high dimensions
arXiv · 2604.15305
The Takeaway
Paul Erdős was a legendary mathematician who thought he knew how points behaved when they were spaced out in high-dimensional grids. He had a specific 'diameter conjecture' that everyone assumed was the rule of the land. These researchers just blew that up, showing that when you get into enough dimensions, the geometry breaks his rules. It is a reminder that our 3D intuition is a terrible guide for the true complexity of reality. This changes how we calculate distances in high-dimensional data, like the kind used to train massive AI models.
From the abstract
Erdős asked whether every $n$-point set in Euclidean space whose $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart must have diameter at least $(1+o(1))n^2$. We disprove this statement by constructing for every prime power $q$ a set $\mathcal X_q\subset \mathbb R^{q^2+q}$ of $n=q+1$ points such that all pairwise distances in $\mathcal X_q$ are mutually at least $1$ apart, while $$\operatorname{diam}(\mathcal X_q)\le\Bigl(1-\frac{1}{\pi^2}+o(1)\Bigr)n^2.$$ The proof is fully forma